Filter function in J

In Matlab the filter is a function that enables convert a LCCDE in time domain to a computable function. That is, for array $a_{n}$ and $b_{n}$ , input signal $x (t)$ , compute $y (t)$ so that the relation below hold.

a_{1} y (t) + a_{2} y (t - 1) + . . . + a_{n} (t - n + 1) = b_{1} x (t) + b_{2} x (t - 1) + . . . + b_{x} (t - n + 1)

As a Functional/Array programmer, it is not difficult to discover the data dependency and get

y (t) = \frac{b \times x (t \dots t - n + 1) - a_{2 \dots n} \times y (t - 1 \dots t - n + 1)}{a_{1}}

and the fact that

b \times x (t \dots t - n + 1)

can be parallelized.

  |comp_b=. [+/@:*(-@i.@#@[) |.!.0"0 1]

The actual y is dependent on the previous n-1 results. This gives the relation to be used by fold (F..):

  |a_1=.{.a
  |a_n=.}.a
  |fn=.],~a_1%~[-a_n+/@:*(#a_n){.]

Finally, assemble all together gives:

  |filter =: {{
  |'a b'=.m
  |comp_b=. [+/@:*(-@i.@#@[) |.!.0"0 1]
  | 
  |a_1=.{.a
  |a_n=.}.a
  |fn=.],~a_1%~[-a_n+/@:*(#a_n){.]
  |(0$0) |.F..fn b comp_b y
  |}}

Bibliography

[filter] 1-D digital filter. Mathworks. https://www.mathworks.com/help/matlab/ref/filter.html.

[nuvoc] NuVoc. J Wiki. https://code.jsoftware.com/wiki/NuVoc.